__tanl.c revision 25c28e83beb90e7c80452a7c818c5e6f73a07dc8
/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
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* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
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*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
/*
* long double __k_tanl(long double x; long double y, int k);
* kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input k indicate -- tan if k=0; else -1/tan
*
* Table look up algorithm
* 1. by tan(-x) = -tan(x), need only to consider positive x
* 2. if x < 5/32 = [0x3ffc4000, 0] = 0.15625 , then
* if x < 2^-57 (hx < 0x3fc40000 0), set w=x with inexact if x != 0
* else
* z = x*x;
* w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
* return (k == 0)? w: 1/w;
* 3. else
* ht = (hx + 0x400)&0x7ffff800 (round x to a break point t)
* lt = 0
* i = (hy-0x3ffc4000)>>11; (i<=64)
* x' = (x - t)+y (|x'| ~<= 2^-7)
* By
* tan(t+x')
* = (tan(t)+tan(x'))/(1-tan(x')tan(t))
* We have
* sin(x')+tan(t)*(tan(t)*sin(x'))
* = tan(t) + ------------------------------- for k=0
* cos(x') - tan(t)*sin(x')
*
* cos(x') - tan(t)*sin(x')
* = - -------------------------------------- for k=1
* tan(t) + tan(t)*(cos(x')-1) + sin(x')
*
*
* where tan(t) is from the table,
* sin(x') = x + pp1*x^3 + ...+ pp5*x^11
* cos(x') = 1 + qq1*x^2 + ...+ qq5*x^10
*/
#include "libm.h"
extern const long double _TBL_tanl_hi[], _TBL_tanl_lo[];
static const long double
one = 1.0L,
/*
* 3 11 -122.32
* |sin(x) - (x+pp1*x +...+ pp5*x )| <= 2 for |x|<1/64
*/
pp1 = -1.666666666666666666666666666586782940810e-0001L,
pp2 = +8.333333333333333333333003723660929317540e-0003L,
pp3 = -1.984126984126984076045903483778337804470e-0004L,
pp4 = +2.755731922361906641319723106210900949413e-0006L,
pp5 = -2.505198398570947019093998469135012057673e-0008L,
/*
* 2 10 -123.84
* |cos(x) - (1+qq1*x +...+ qq5*x )| <= 2 for |x|<=1/128
*/
qq1 = -4.999999999999999999999999999999378373641e-0001L,
qq2 = +4.166666666666666666666665478399327703130e-0002L,
qq3 = -1.388888888888888888058211230618051613494e-0003L,
qq4 = +2.480158730156105377771585658905303111866e-0005L,
qq5 = -2.755728099762526325736488376695157008736e-0007L,
/*
* |tan(x) - (x+t1*x^3+...+t6*x^13)|
* |------------------------------ | <= 2^-59.73 for |x|<0.15625
* | x |
*/
t1 = +3.333333333333333333333333333333423342490e-0001L,
t2 = +1.333333333333333333333333333093838744537e-0001L,
t3 = +5.396825396825396825396827906318682662250e-0002L,
t4 = +2.186948853615520282185576976994418486911e-0002L,
t5 = +8.863235529902196573354554519991152936246e-0003L,
t6 = +3.592128036572480064652191427543994878790e-0003L,
t7 = +1.455834387051455257856833807581901305474e-0003L,
t8 = +5.900274409318599857829983256201725587477e-0004L,
t9 = +2.391291152117265181501116961901122362937e-0004L,
t10 = +9.691533169382729742394024173194981882375e-0005L,
t11 = +3.927994733186415603228178184225780859951e-0005L,
t12 = +1.588300018848323824227640064883334101288e-0005L,
t13 = +6.916271223396808311166202285131722231723e-0006L;
#define i0 0
long double
__k_tanl(long double x, long double y, int k) {
long double a, t, z, w = 0, s, c;
int *pt = (int *) &t, *px = (int *) &x;
int i, j, hx, ix;
t = 1.0L;
hx = px[i0];
ix = hx & 0x7fffffff;
if (ix < 0x3ffc4000) {
*(3 - i0 + (int *) &t) = 1; /* make t = one+ulp */
if (ix < 0x3fc60000) {
if (((int) (x * t)) < 1) /* generate inexact */
w = x; /* generate underflow if subnormal */
} else {
z = x * x;
if (ix < 0x3ff30000) /* 2**-12 */
t = z * (t1 + z * (t2 + z * (t3 + z * t4)));
else
t = z * (t1 + z * (t2 + z * (t3 + z * (t4 +
z * (t5 + z * (t6 + z * (t7 + z * (t8 +
z * (t9 + z * (t10 + z * (t11 +
z * (t12 + z * t13))))))))))));
t = y + x * t;
w = x + t;
}
return (k == 0 ? w : -one / w);
}
j = (ix + 0x400) & 0x7ffff800;
i = (j - 0x3ffc4000) >> 11;
pt[i0] = j;
if (hx > 0)
x = y - (t - x);
else
x = (-y) - (t + x);
a = _TBL_tanl_hi[i];
z = x * x;
/* cos(x)-1 */
t = z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
/* sin(x) */
s = x * (one + z * (pp1 + z * (pp2 + z * (pp3 + z * (pp4 + z * pp5)))));
if (k == 0) {
w = a * s;
t = _TBL_tanl_lo[i] + (s + a * w) / (one - (w - t));
return (hx < 0 ? -a - t : a + t);
} else {
w = s + a * t;
c = w + _TBL_tanl_lo[i];
z = one - (a * s - t);
return (hx >= 0 ? z / (-a - c) : z / (a + c));
}
}